The BTS reference guide indicates that certain leaders have favorite civics and some have favorite religions. I've often wondered just how that plays out in a practical sense.

Is the only value in having that knowledge, in that it gives the human player a clue as to what the AI will do? Or is there any extra benefit when the human player adopts a favored civic/religion when playing as a given leader?

If you are both in that civic, you get a bonus similar to that for shared religion. Different leaders weight religion vs favorite civics differently.

It also causes a bug that for some reason some people believe was intended, where you can't bribe a leader in their favorite civic to any other civic, no matter the civic category. So yes, this means you can't bribe Hatty (who likes Organized Religion) into representation because it goes against everything she stands for because she is in OR. Should she go FR, then you could bribe her again. I think it was solver who believed this was intended, but I believe it's utter nonsense.

Anyway, use favorite civics for diplo -----> these are often a top way to garner UN votes especially if in an otherwise religiously turbulent world you need to run FR. Also note that you can flip civics to force AIs into older civics to retain the bonus for this purpose. A transport (or even just a galleon) full of spies that you can move in, run the mission, and get out is very effective for doing that, and keeps the spy diplo hit risk at a minimum.

I think it was solver who believed this was intended, but I believe it's utter nonsense.
I fully agree.
The message should only refer to civics in the same catagory and not block
all 5 civic catagories.

Ditto again to what TMIT said. Its a bug that you cant change other civics.

I have a question about this part of your query:
The BTS reference guide indicates that certain leaders have. . . favorite religions.
Does this mean that certain AIs have a tendency to found certain religions? I know Izzy like Buddhism, Sal likes Hinduism, etc, but I dont think this is the same game mechanic as Favorite Civic. For example, lets say Sal doesnt found Hinduism, and grabs Monotheism instead, I cant make him change to Hinduism without some serious missionary building (and even then, since Sal's civic is Theocracy).

"Favored" religions may also come into play if the "Choose Religion" option is checked. Sal my ALWAYS pick Hinduism if its available when he is the first to research a tech that founds a religion, although this should be (and may be) Islam. I dont use the "choose Religion" option much though, so I cant say for sure if that is another aspect of "Favorite Religion".

Favoured religion means the religion the AI will choose to found if the choose-your-religion box is checked. Most of these are simple historical things, Izzy will always take Christianity, Gandhi Hinduism, etc. You may still see a Muslim Justinian, but that is because some other AI took Christianity. Some AIs appear to choose religions purely randomly, but the unfavoured ones tend to be Judiasm and the two Chinese ones (it may just be that I never let Mao, QSH, or Toko found their own).

There was info about a leader's favourite religion in the leader's profile menu.
It was introduced in Warlords or early BTS, but for some reason it has been removed again.

I like to play with 'choose religion' option, but it spoils which leaders could be in the game.

Some examples :
Buddhism -> Asoka, Suryavarman, Tokugawa
Hinduism -> Gandhi
Islam -> Saladin, Ottoman leaders, Mansa Musa
Confucianism -> Wang Kon
Christianity -> English leaders, Isabella, Charlemagne, Justinian

add Zara to Christianity :p

Some AIs don't have a favorite religion -----> they just select the default religion for the tech in that case.

Here's a full list I compiled for you from the Sevopedia.

Christianity: Bismarck, Catherine, Charlemagne, Churchill, De Gaulle, Elizabeth, Frederick, Isabella, Joao II, Justinian I, Lincoln, Louis XIV, Napoleon, Peter, Roosevelt, Victoria, Washington, Willem van Oranje, Zara Yaqob
Islam: Mansa Musa, Mehmed II, Saladin, Suleiman
Buddhism: Asoka, Kublai Khan, Suryavarman II, Tokugawa
Hinduism: Ghandi
Confuscianism: Wang Kon
Taoism: Qin Shi Huang
Judaism: None
Filthy Pagans :devil: : Alexander, Augustus Caesar, Boudica, Brennus, Cyrus, Darius I, Genghis Khan, Gilgamesh, Hammurabi, Hannibal, Hatshepsut, Huayna Capac, Julius Caesar, Mao Zedong, Montezuma, Pacal II, Pericles, Ragnar, Ramesses II, Shaka, Sitting Bull, Stalin